I'm leaning towards the line “Thus conscience does make cowards of us all” (III.i.83) when Hamlet speaks of the cowardice that people have when facing the unknown of death, but I'm not really sure how it's essential or integral to the play. Perhaps it hints at one of the play's themes?
Post #1 has a good quote that could address the theme of responsibility. Hamlet speaks of cowardice, and there are moments in the play where he has to take action and responsibility toward Claudius or his relationship with Ophelia. "The conscience" or the idea of thinking about one's actions is certainly prevalent, mostly because Hamlet does too much thinking and not enough action.
Conscience is a huge theme in the play and this quote reflects that excelently. It describes what happens to Hamlet. In the heat of the moment Hamlet is never afraid to act. Yet given a moment to think about it and the prominent moral code which we see from Hamlet makes him a coward to whatever deed he had been about to perform.
"conscience doth make cowards of us all"
to an Elizabethan audience:
conscience = "moral scrupple" and "reflectiveness"
Hamlet is either too moral, or too reflective, which leads to his lethargy and inability to act on Claudius until it's too late. This passage is a possible discussion of his hamartia.
Then again, his fatal flaw could also be his melancholy. As discussed quite literally in this soliloquy, Hamlet is depressed and wants to die.
This line addresses the central conflict of the play: how to pursue
justice in a fallen world without losing your moral standing.
The reason Hamlet is afraid of death is that, in Christian terms, some
punishment might await those who commit sinful acts. Earlier in the soliloquy, he contemplates suicide, but Christian doctrine prohibits suicide, as it does murder and vengeance. Hamlet wants to avenge his father's murder, but he wants to avoid committing any kind of wrong action--or in the play's Christian
framework, "sin"--that might lead to punishment and torment in the
afterlife, if there is one. Elsewhere in the play, Claudius and the ghost of Hamlet's father also talk about the themes of judgment and punishment in the hereafter.
I don't think Hamlet is a religious play necessarily, but I do think that it contains religious ideas, in this case, Christian ideas.
In plainer, nonreligious terms, Hamlet has a dirty job to do, but he wants to be able to wash his hands after doing it. The tragedy of the play is that he can't have it both ways.